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Theorem cnextf 21081
Description: Extension by continuity. The extension by continuity is a function. (Contributed by Thierry Arnoux, 25-Dec-2017.)
Hypotheses
Ref Expression
cnextf.1  |-  C  = 
U. J
cnextf.2  |-  B  = 
U. K
cnextf.3  |-  ( ph  ->  J  e.  Top )
cnextf.4  |-  ( ph  ->  K  e.  Haus )
cnextf.5  |-  ( ph  ->  F : A --> B )
cnextf.a  |-  ( ph  ->  A  C_  C )
cnextf.6  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  C )
cnextf.7  |-  ( (
ph  /\  x  e.  C )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  =/=  (/) )
Assertion
Ref Expression
cnextf  |-  ( ph  ->  ( ( JCnExt K
) `  F ) : C --> B )
Distinct variable groups:    x, A    x, B    x, C    x, F    x, J    x, K    ph, x

Proof of Theorem cnextf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cnextf.3 . . . 4  |-  ( ph  ->  J  e.  Top )
2 cnextf.4 . . . 4  |-  ( ph  ->  K  e.  Haus )
3 cnextf.5 . . . 4  |-  ( ph  ->  F : A --> B )
4 cnextf.a . . . 4  |-  ( ph  ->  A  C_  C )
5 cnextf.1 . . . . 5  |-  C  = 
U. J
6 cnextf.2 . . . . 5  |-  B  = 
U. K
75, 6cnextfun 21079 . . . 4  |-  ( ( ( J  e.  Top  /\  K  e.  Haus )  /\  ( F : A --> B  /\  A  C_  C
) )  ->  Fun  ( ( JCnExt K
) `  F )
)
81, 2, 3, 4, 7syl22anc 1269 . . 3  |-  ( ph  ->  Fun  ( ( JCnExt
K ) `  F
) )
9 simpl 459 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  ph )
10 cnextf.6 . . . . . . . . 9  |-  ( ph  ->  ( ( cls `  J
) `  A )  =  C )
1110eleq2d 2514 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ( ( cls `  J
) `  A )  <->  x  e.  C ) )
1211biimpar 488 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  x  e.  ( ( cls `  J
) `  A )
)
13 cnextf.7 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  =/=  (/) )
14 n0 3741 . . . . . . . 8  |-  ( ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  =/=  (/)  <->  E. y 
y  e.  ( ( K  fLimf  ( (
( nei `  J
) `  { x } )t  A ) ) `  F ) )
1513, 14sylib 200 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  E. y 
y  e.  ( ( K  fLimf  ( (
( nei `  J
) `  { x } )t  A ) ) `  F ) )
16 haustop 20347 . . . . . . . . . . . . . 14  |-  ( K  e.  Haus  ->  K  e. 
Top )
172, 16syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  Top )
185, 6cnextfval 21077 . . . . . . . . . . . . 13  |-  ( ( ( J  e.  Top  /\  K  e.  Top )  /\  ( F : A --> B  /\  A  C_  C
) )  ->  (
( JCnExt K ) `
 F )  = 
U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
191, 17, 3, 4, 18syl22anc 1269 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( JCnExt K
) `  F )  =  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
2019eleq2d 2514 . . . . . . . . . . 11  |-  ( ph  ->  ( <. x ,  y
>.  e.  ( ( JCnExt
K ) `  F
)  <->  <. x ,  y
>.  e.  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) ) )
21 opeliunxp 4886 . . . . . . . . . . 11  |-  ( <.
x ,  y >.  e.  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  <->  ( x  e.  ( ( cls `  J
) `  A )  /\  y  e.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
2220, 21syl6bb 265 . . . . . . . . . 10  |-  ( ph  ->  ( <. x ,  y
>.  e.  ( ( JCnExt
K ) `  F
)  <->  ( x  e.  ( ( cls `  J
) `  A )  /\  y  e.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) ) )
2322exbidv 1768 . . . . . . . . 9  |-  ( ph  ->  ( E. y <.
x ,  y >.  e.  ( ( JCnExt K
) `  F )  <->  E. y ( x  e.  ( ( cls `  J
) `  A )  /\  y  e.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) ) )
24 19.42v 1834 . . . . . . . . 9  |-  ( E. y ( x  e.  ( ( cls `  J
) `  A )  /\  y  e.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  <->  ( x  e.  ( ( cls `  J
) `  A )  /\  E. y  y  e.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
2523, 24syl6bb 265 . . . . . . . 8  |-  ( ph  ->  ( E. y <.
x ,  y >.  e.  ( ( JCnExt K
) `  F )  <->  ( x  e.  ( ( cls `  J ) `
 A )  /\  E. y  y  e.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) ) )
2625biimpar 488 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( ( cls `  J
) `  A )  /\  E. y  y  e.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )  ->  E. y <. x ,  y >.  e.  ( ( JCnExt K ) `
 F ) )
279, 12, 15, 26syl12anc 1266 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  E. y <. x ,  y >.  e.  ( ( JCnExt K
) `  F )
)
2825simprbda 629 . . . . . . 7  |-  ( (
ph  /\  E. y <. x ,  y >.  e.  ( ( JCnExt K
) `  F )
)  ->  x  e.  ( ( cls `  J
) `  A )
)
2911adantr 467 . . . . . . 7  |-  ( (
ph  /\  E. y <. x ,  y >.  e.  ( ( JCnExt K
) `  F )
)  ->  ( x  e.  ( ( cls `  J
) `  A )  <->  x  e.  C ) )
3028, 29mpbid 214 . . . . . 6  |-  ( (
ph  /\  E. y <. x ,  y >.  e.  ( ( JCnExt K
) `  F )
)  ->  x  e.  C )
3127, 30impbida 843 . . . . 5  |-  ( ph  ->  ( x  e.  C  <->  E. y <. x ,  y
>.  e.  ( ( JCnExt
K ) `  F
) ) )
3231abbi2dv 2570 . . . 4  |-  ( ph  ->  C  =  { x  |  E. y <. x ,  y >.  e.  ( ( JCnExt K ) `
 F ) } )
33 dfdm3 5022 . . . 4  |-  dom  (
( JCnExt K ) `
 F )  =  { x  |  E. y <. x ,  y
>.  e.  ( ( JCnExt
K ) `  F
) }
3432, 33syl6reqr 2504 . . 3  |-  ( ph  ->  dom  ( ( JCnExt
K ) `  F
)  =  C )
35 df-fn 5585 . . 3  |-  ( ( ( JCnExt K ) `
 F )  Fn  C  <->  ( Fun  (
( JCnExt K ) `
 F )  /\  dom  ( ( JCnExt K
) `  F )  =  C ) )
368, 34, 35sylanbrc 670 . 2  |-  ( ph  ->  ( ( JCnExt K
) `  F )  Fn  C )
3719rneqd 5062 . . 3  |-  ( ph  ->  ran  ( ( JCnExt
K ) `  F
)  =  ran  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) ) )
38 rniun 5246 . . . 4  |-  ran  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  = 
U_ x  e.  ( ( cls `  J
) `  A ) ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )
39 vex 3048 . . . . . . . . 9  |-  x  e. 
_V
4039snnz 4090 . . . . . . . 8  |-  { x }  =/=  (/)
41 rnxp 5267 . . . . . . . 8  |-  ( { x }  =/=  (/)  ->  ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  =  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )
4240, 41ax-mp 5 . . . . . . 7  |-  ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  =  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )
4311biimpa 487 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ( cls `  J
) `  A )
)  ->  x  e.  C )
446toptopon 19948 . . . . . . . . . . 11  |-  ( K  e.  Top  <->  K  e.  (TopOn `  B ) )
4517, 44sylib 200 . . . . . . . . . 10  |-  ( ph  ->  K  e.  (TopOn `  B ) )
4645adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  C )  ->  K  e.  (TopOn `  B )
)
475toptopon 19948 . . . . . . . . . . . 12  |-  ( J  e.  Top  <->  J  e.  (TopOn `  C ) )
481, 47sylib 200 . . . . . . . . . . 11  |-  ( ph  ->  J  e.  (TopOn `  C ) )
4948adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  C )  ->  J  e.  (TopOn `  C )
)
504adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  C )  ->  A  C_  C )
51 simpr 463 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  C )  ->  x  e.  C )
52 trnei 20907 . . . . . . . . . . 11  |-  ( ( J  e.  (TopOn `  C )  /\  A  C_  C  /\  x  e.  C )  ->  (
x  e.  ( ( cls `  J ) `
 A )  <->  ( (
( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) ) )
5352biimpa 487 . . . . . . . . . 10  |-  ( ( ( J  e.  (TopOn `  C )  /\  A  C_  C  /\  x  e.  C )  /\  x  e.  ( ( cls `  J
) `  A )
)  ->  ( (
( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) )
5449, 50, 51, 12, 53syl31anc 1271 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  C )  ->  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A ) )
553adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  C )  ->  F : A --> B )
56 flfelbas 21009 . . . . . . . . . . 11  |-  ( ( ( K  e.  (TopOn `  B )  /\  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A )  /\  F : A --> B )  /\  y  e.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  -> 
y  e.  B )
5756ex 436 . . . . . . . . . 10  |-  ( ( K  e.  (TopOn `  B )  /\  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A )  /\  F : A --> B )  ->  (
y  e.  ( ( K  fLimf  ( (
( nei `  J
) `  { x } )t  A ) ) `  F )  ->  y  e.  B ) )
5857ssrdv 3438 . . . . . . . . 9  |-  ( ( K  e.  (TopOn `  B )  /\  (
( ( nei `  J
) `  { x } )t  A )  e.  ( Fil `  A )  /\  F : A --> B )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  C_  B
)
5946, 54, 55, 58syl3anc 1268 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F )  C_  B
)
6043, 59syldan 473 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ( cls `  J
) `  A )
)  ->  ( ( K  fLimf  ( ( ( nei `  J ) `
 { x }
)t 
A ) ) `  F )  C_  B
)
6142, 60syl5eqss 3476 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ( cls `  J
) `  A )
)  ->  ran  ( { x }  X.  (
( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B )
6261ralrimiva 2802 . . . . 5  |-  ( ph  ->  A. x  e.  ( ( cls `  J
) `  A ) ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B )
63 iunss 4319 . . . . 5  |-  ( U_ x  e.  ( ( cls `  J ) `  A ) ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B 
<-> 
A. x  e.  ( ( cls `  J
) `  A ) ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B )
6462, 63sylibr 216 . . . 4  |-  ( ph  ->  U_ x  e.  ( ( cls `  J
) `  A ) ran  ( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B )
6538, 64syl5eqss 3476 . . 3  |-  ( ph  ->  ran  U_ x  e.  ( ( cls `  J
) `  A )
( { x }  X.  ( ( K  fLimf  ( ( ( nei `  J
) `  { x } )t  A ) ) `  F ) )  C_  B )
6637, 65eqsstrd 3466 . 2  |-  ( ph  ->  ran  ( ( JCnExt
K ) `  F
)  C_  B )
67 df-f 5586 . 2  |-  ( ( ( JCnExt K ) `
 F ) : C --> B  <->  ( (
( JCnExt K ) `
 F )  Fn  C  /\  ran  (
( JCnExt K ) `
 F )  C_  B ) )
6836, 66, 67sylanbrc 670 1  |-  ( ph  ->  ( ( JCnExt K
) `  F ) : C --> B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444   E.wex 1663    e. wcel 1887   {cab 2437    =/= wne 2622   A.wral 2737    C_ wss 3404   (/)c0 3731   {csn 3968   <.cop 3974   U.cuni 4198   U_ciun 4278    X. cxp 4832   dom cdm 4834   ran crn 4835   Fun wfun 5576    Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   ↾t crest 15319   Topctop 19917  TopOnctopon 19918   clsccl 20033   neicnei 20113   Hauscha 20324   Filcfil 20860    fLimf cflf 20950  CnExtccnext 21074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-map 7474  df-pm 7475  df-rest 15321  df-fbas 18967  df-fg 18968  df-top 19921  df-topon 19923  df-cld 20034  df-ntr 20035  df-cls 20036  df-nei 20114  df-haus 20331  df-fil 20861  df-fm 20953  df-flim 20954  df-flf 20955  df-cnext 21075
This theorem is referenced by:  cnextcn  21082  cnextfres1  21083
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